Question

Please answer 2 and 3. I have the answer for 1.

A firm's total cost function is given by the equation: TC = 4000 + 5Q + 10Q2.

(1) Write an expression for each of the following cost concepts:

a. Total Fixed Cost

b. Average Fixed Cost

c. Total Variable Cost

d. Average Variable Cost

e. Average Total Cost

f. Marginal Cost

(2) Determine the quantity that minimizes average total cost and minimizing average variable cost.

(3) Why does its average variable cost curve achieve its minimum at a lower level of output than the average total cost curve?

Answer 1

Solution -

(2) Determine the quantity that minimizes average total cost and minimizing average variable cost.

ATC is minimized where MC is equal to ATC.

Equating MC to ATC

4000 + 5Q + 10Q2 / Q= 5 + 20Q

4000 +5Q + 102= 5Q + 20Q2

4000 = 10Q2

Q2= 400

Q = 20

ATC is minimized at 20 units of output. Up to 20, ATC falls, while beyond 20 ATC rises. MC should be less than ATC for any quantity less than 20. For example, let Q = 10: MC = 5 + 20(10) = 205

TC = 4000* 5(10)*10(10)^2 / 10= 505

MC is indeed less than ATC for quantities smaller than 20.

why does its average variable cost curve achieve its minimum at a lower level of output than the average total cost curve?

Answer: The average total cost is still decreasing when the average variable cost begins to increase. This is because the average total cost equals average variable cost plus F C/q, which is decreasing everywhere. Therefore the minimum of the average variable cost lies to the left of the minimum of average total cost.

Another way to answer: The derivative of the average variable cost is d(V C(q)/q)/dq = (qV C0 (q) ? V C(q))/q2 . Setting it equal to zero, we get qMC(q) = V C(q). The derivative of the average total cost curve is d((V C(q) + F C)/q)/dq = (qMC(q) ? (V C(q) + F C))/q2 . Setting it equal to zero, we get qMC(q) = V C(q) + F C. Since V C(q) + F C > V C(q), and marginal costs are increasing at a minimum of both AV C and AT C, the solution to qMC(q) = V C(q) + F C must be greater than the solution to qMC(q) = V C(q).